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A081568 Third binomial transform of Fibonacci(n+1). 7
1, 4, 17, 75, 338, 1541, 7069, 32532, 149965, 691903, 3193706, 14745009, 68084297, 314394980, 1451837593, 6704518371, 30961415074, 142980203437, 660285858245, 3049218769908, 14081386948661, 65028302171639, 300302858766202 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A081567.

Case k=3 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.

a(n) = 4^n*a(n;1/4) = Sum_{k=0..n} binomial(n,k)(-1)^k F(k-1) 4^(n-k), which also implies Deléham's formula given below and where a(n;d), n=0,1,..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012

REFERENCES

D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted to Fibonacci Quart. 2012).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042

Index entries for linear recurrences with constant coefficients, signature (7,-11).

FORMULA

a(n) = 7*a(n-1) - 11*a(n-2), a(0)=1, a(1)=4.

a(n) = (1/2 - sqrt(5)/10)*(7/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 7/2)^n = A099453(n)-3*A099453(n-1).

G.f.: (1-3*x)/(1-7*x+11*x^2).

a(n) = Sum_{k=0..n} A094441(n,k)*3^k. - Philippe Deléham, Dec 14 2009

G.f.: Q(0,u)/x -1/x, where u=x/(1-3*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k+1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013

MATHEMATICA

CoefficientList[Series[(1 - 3 x) / (1 - 7 x + 11 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

LinearRecurrence[{7, -11}, {1, 4}, 30] (* Harvey P. Dale, Feb 01 2015 *)

PROG

(MAGMA) I:=[1, 4]; [n le 2 select I[n] else 7*Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

(PARI) Vec((1-3*x)/(1-7*x+11*x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

CROSSREFS

Cf. A000045, A161731 (INVERT transform), A007582 (INVERTi transform), A081569 (binomial transform).

Sequence in context: A227504 A218984 A289800 * A026378 A265680 A255714

Adjacent sequences:  A081565 A081566 A081567 * A081569 A081570 A081571

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 22 2003

STATUS

approved

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Last modified February 19 08:57 EST 2019. Contains 320309 sequences. (Running on oeis4.)